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Bicek was drafted 131st overall by the New Jersey Devils in the 1997 NHL Entry Draft. He then proceeded to play 7 pro seasons in North America, mostly playing for the Albany River Rats of the AHL. In 423 games in Albany, Bicek scored 75 goals and 167 assists for 242 points. In those last four seasons, he was called up to the Devils sporadically. During the 2004–05 NHL lockout, Bicek returned to Slovakia to play for HC Košice. He played 54 games there and scored 18 goals and 41 points. In 62 career NHL games, Bicek has scored 6 goals and 7 assists for 13 points. He won a Stanley Cup in 2003 with the Devils.